There is a challenging problem in a book of mine on complex analysis, and I seriously do not even know where to start. I'm more than sure I don't properly understand the problem.
Prove that there is a constant $C$, independent of $n$, such that if $\{z_j\}$ are complex numbers and if $$\sum\limits_{j=1}^n |z_j| \geq 1,$$ then there is a subcollection $\{z_{j_1},...,z_{j_k}\}\subseteq\{z_1,...,z_n\}$ such that $$ \Big |\sum\limits_{m=1}^k z_{j_m}\Big | \geq C.$$ Can you find the best constant C? [Suggestion: Look at the arguments of the nonzero $z_j$. At least one third of these arguments lie within $\frac{2\pi}{3}$ of each other.]
Clearly $C=0$ is such a constant that will always satisfy those two inequalities independent of $n$ given any subset of an arbitrary $\{z_j\}$, and any subset of $\{z_j\}$ that doesn't go back to the origin will add up to some random number $C$, but of course that's too simple (not what the problem is looking for I'm sure). Why does the first inequality matter.
I also have no idea what is meant by 'the best constant $C$'. What does 'best' mean?
I can see what the suggestion means, and I have a feeling as to what is could be used for, but I need to understand the problem first.
Please answer my questions and give me some guidance so I can understand this problem, but don't give me any answers (please).